L^pestimates for the Hilbert transforms along a one-variable vector field

Abstract

Abstract. Stein conjectured that the Hilbert transform in the direction of a vector field is bounded on, say, L 2 whenever v is Lipschitz. We establish a wide range of L p estimates for this operator when v is a measurable, non-vanishing, one-variable vector field in R 2 . Aside from an L 2 estimate following from a simple trick with Carleson's theorem, these estimates were unknown previously. This paper is closely related to a recent paper of the first author ([2]).

Hilbert transform along measurable vector fields constant on Lipschitz curves: $L^p$ boundedness

“…After excluding the exceptional sets, the argument in [5], together with [4](which also works equally well for our case as has been pointed out in [15]), will lead to the square function estimate, i.e. Proposition 3.1.…”

“…However, we have one more assumption that u ∞ ≤ 1. To recover the result in Theorem 1.2, we just need to apply the following unisotropic scaling 5) and a simple limiting argument.…”

“…When trying to put all the frequency annuli together to prove the boundedness of the whole operator H v , we have increased the complexity of the problem "by one dimension". Fortunately, this can be compensated by making use of the Symmetry B, which is done by Bateman and Thiele in [5] through a square function estimate.…”

A guide to Carleson's theorem

“…It uses both mass and size and thus it combines elements of the two proofs we have seen in earlier sections. The interplay between mass and size made this approach particularly well suited for applications to the problem of singular integrals along vector fields [15], [1], [2], since it opened the door to the use of Kakeya type maximal functions.…”

“…Theorem 1.1 has since received many proofs, most notably by Fefferman [10] and by Lacey and Thiele [13]. The impact of Carleson's Theorem has increased in recent years thanks to its connections with Scattering Theory [19], Egodic Theory [7], [8], the theory of directional singular integrals in the plane [15], [16], [6], [9], [1], [2] and the theory of operators with quadratic modulations [17], [18]. A more detailed description can be found in [12].…”